Optimal seatings changes 3R+F

Please disregard those results, for some of them are wrong due to a flaw in the validation.

From now on, I strongly recommand not to use the version 1.5e but version 1.5f :

Excel: www.vekn.net/images/stories/downloads/thearchon1.5f.xlsx
LibreOffice: www.vekn.net/images/stories/downloads/thearchon1.5f.ods

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Ok, I've started toying with the 3R seatings.

Starting with 10 players, the current seating is :

1	2	3	4	5	|	6	7	8	9	10
10	1	9	8	2	|	5	6	4	3	7
4	7	10	3	1	|	9	2	5	8	6

with the following results:

Rule	Expected	Result
N		10
1. No pair of players repeat their predator-prey relationship. This is mandatory, by the VEKN rules.		OK
2. No pair of players share a table through all two rounds, when possible. (N/A in some 2R event.)		KO (player 1 shares a table with player 2 on round 2)
3. Available VPs are equitably distributed.	OK	OK
5. A player doesn't sit in the fifth seat more than once.		OK
6. No pair of players repeat the same relative position[*], when possible.		KO (players 8 and 10, table 1, round 2)
7. A player doesn't play in the same seat position, if possible.		KO (player 7, table 1, position 2)
8. Starting transfers are equitably distributed. [NOAL]		KO. Absolute deviation is: 1,04 => 1, 6 have 7 transfers | 2, 4, 5, 7, 9, 10 have 8 transfers | 3, 8 have 11 transfers

The first of the best results I have computed is:

1	2	3	4	5	|	6	7	8	9	10
8	6	2	10	1	|	5	9	4	7	3
10	3	9	2	8	|	4	1	7	5	6

with the following results:

Rule	Expected	Result
N		10
1. No pair of players repeat their predator-prey relationship. This is mandatory, by the VEKN rules.		OK
2. No pair of players share a table through all two rounds, when possible. (N/A in some 2R event.)		KO (player 1 shares a table with player 2 on round 2)
3. Available VPs are equitably distributed.	OK	OK
5. A player doesn't sit in the fifth seat more than once.		OK
6. No pair of players repeat the same relative position[*], when possible.		OK
7. A player doesn't play in the same seat position, if possible.		OK
8. Starting transfers are equitably distributed. [NOAL]		KO. Absolute deviation is: 0,72 => 1, 6 have 7 transfers | 2, 3, 5, 7, 9, 10 have 9 transfers | 4, 8 have 8 transfers

So compared to the current seating, rule 2 and 7 are no longer violated and the distribution of the transfers has a lesser deviation.
Plus, everyone has at least once 4 transfers.

Do you see anything unoptimal with the proposed new seating?

8 players (3R): unchanged 7 1 8 2 | 3 5 4 6 / 4 7 6 1 | 8 3 2 5
Rule 2 KO (player 1 shares a table with player 2 on round 2): 12 (once), 12 (twice)
Rule 8 KO. Absolute deviation is: 1 => 1, 5 have 7 transfers | 2, 6 have 9 transfers | 3, 7 have 6 transfers | 4, 8 have 8 transfers

9 players (3R): unchanged 9 4 2 8 6 | 5 3 7 1 / 8 1 9 7 3 | 4 6 5 2
Rule 2 KO (player 1 shares a table with player 3 on round 2): 15 (once), 12 (twice), 3 (thrice)
Rule 3 KO. Absolute deviation is: 0,222 => 1, 2, 3, 4, 8, 9 have 14 VP | 5, 6, 7 have 13 VP
Rule 8 KO. Absolute deviation is: 0,666 => 1, 4, 6 have 7 transfers | 2, 3, 7 have 9 transfers | 5, 8, 9 have 8 transfers

10 players (3R): unchanged 10 1 9 8 2 | 5 6 4 3 7 / 4 7 10 3 1 | 9 2 5 8 6
Rule 2 KO. (player 1 shares a table with player 2 on round 2): 18 (once), 18 (twice), 2 (thrice))
Rule 6 and 7 KO.
Rule 8 KO. Absolute transfer deviation is: 1,04
The alternative seating (8 6 4 1 3 | 9 5 2 10 7 / 10 4 9 6 1 | 5 3 7 2 8 ) will not be kept because even though it doesn't violate rule 6 and 7 and has a better transfer deviation, it violates rule 2 with 3 pairs of players meeting thrice.

12 players (3R): 6 9 5 1 | 11 8 10 3 | 2 4 12 7 / 8 1 9 2 | 7 5 4 11 | 12 3 6 10
Rule 2 KO (player 2 shares a table with player 4 on round 2): 36 (once), 9 (twice)
(Current seating violates rule 2 (player 1 shares a table with player 2 on round 2): 30 (once), 12 (twice))
Rule 8 KO. Absolute deviation is: 1 => 1, 5, 9 have 7 transfers | 2, 6, 10 have 9 transfers | 3, 7, 11 have 6 transfers | 4, 8, 12 have 8 transfers

13 players (3R): 12 10 1 6 8 | 11 4 2 7 | 13 5 9 3 / 9 13 7 1 11 | 5 6 10 2 | 3 8 4 12
Rule 2 KO (player 2 shares a table with player 4 on round 2): 42 (once), 12 (twice)
(Current seating violates rule 2 (player 1 shares a table with player 3 on round 2): 34 (once), 16 (twice))
Rule 3 KO. Absolute deviation is: 0,071 => 1 have 15 VP | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 have 13 VP
(Current seating has the same absolute vp deviation)
Rule 8 KO. Absolute deviation is: 0,804 => 1, 3, 9, 12 have 8 transfers | 2, 4, 7, 8 have 9 transfers | 5, 6, 11, 13 have 7 transfers | 10 have 6 transfers
(Current seating has the same absolute transfer deviation)

14 players (3R): 12 6 1 11 7 | 14 10 2 13 3 | 4 9 5 8 / 5 11 9 6 14 | 8 13 4 12 2 | 7 3 10 1
Rule 2 KO (player 2 shares a table with player 3 on round 2): 42 (once), 18 (twice)
(Current seating has the same encounter repartition)
Rule 3 KO. Absolute deviation is: 0,0612 => 1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14 have 14 VP | 2, 6 have 15 VP
(Current seating has the same absolute deviation)
(Current seating violates rule 7: player 12, table 2, position 2)
Rule 8 KO. Absolute deviation is: 0,734 => 1, 4, 5, 8 have 8 transfers | 2, 3, 9, 10, 13, 14 have 9 transfers | 6, 7, 11, 12 have 7 transfers
(Current seating has an absolute deviation of 0,755)

15 players (3R): 5 11 6 1 12 | 13 4 7 2 8 | 9 3 15 10 14 / 14 1 4 11 7 | 12 15 2 6 9 | 10 8 5 13 3 (fixed in 1.5f)
Rule 2 KO (player 1 shares a table with player 5 on round 2): 54 (once), 18 (twice)
(Current seating has the same encounter repartition)
(Current seating also violate rule 6 and 7: player 8, table 1, position 3)
Rule 8 KO. Absolute deviation is: 0,72 => 1, 11, 12 have 7 transfers | 2, 3, 4, 7, 8, 9, 10, 14, 15 have 9 transfers | 5, 6, 13 have 8 transfers
(Current seating has an absolute deviation of 1,12)

16 players (3R): unchanged 8 1 14 11 | 12 5 2 15 | 16 9 6 3 | 4 13 10 7 / 15 7 1 9 | 10 16 8 2 | 3 11 13 5 | 6 4 12 14
Rule 8 KO.
(Current seating has an absolute transfer deviation of 1)

17 players (3R) (in progress) (fixed in 1.5f)

For 17 players, I had to "cheat" a little by entering manually some seatings (otherwise it would take 1 year to compute), so I don't know if there is a better solution. Anyway, this seating is slightly better.
What I did is that I chose 2 players that will never be seated on the only 5-players table (since only 15 players can be seated there through the 3 rounds). I chose players 12 and 17. I seated them in 4th, 3rd, 2nd position so they get in exchange the most transfers. The other players can only be seated once on the 5-players table, and I limited the number of players with 6 transfers to two.
Players 1 and 14 are 1st, 2nd and 3rd (6 transfers) but some players are necessarily disadvantaged.

EDIT:
The seating in the version 1.5d of the archon had two pairs of players meeting each other in a prey-predator then predator-prey way.
EDIT: the seating in version 1.5a had no violation of rule 1, therefore we use it until we found a better seating: 17 13 11 6 8 | 3 1 7 15 | 5 9 2 12 | 4 10 14 16 / 15 12 9 7 10 | 11 3 6 14 | 16 8 1 5 | 13 17 4 2

18 players (3R) 12 15 11 6 1 | 16 18 7 2 13 | 17 4 14 8 | 9 5 10 3 / 13 8 5 15 17 | 3 14 9 11 16 | 2 6 18 12 | 10 1 4 7
Rule 2 KO (player 3 shares a table with player 5 on round 2): 72 (once), 12 (twice)
(Current seating has a worse encounter repartition: 66 (once), 15 (twice))
Rule 3 KO. Absolute deviation is: 0,222 => 1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 15, 16 have 14 VP | 4, 10, 12, 14, 17, 18 have 13 VP
(Current seating has the same absolute VP deviation.)
Rule 8 KO. Absolute deviation is: 0,666 => 1, 2, 6, 12, 15, 16 have 7 transfers | 3, 9, 10, 11, 13, 17 have 8 transfers | 4, 5, 7, 8, 14, 18 have 9 transfers
(Current seating has an absolute transfer deviation of 0,777)

19 players (3R): 12 4 11 6 16 | 17 14 7 18 1 | 2 19 9 13 8 | 5 10 15 3 / 13 3 10 17 11 | 18 5 19 12 6 | 15 8 1 16 7 | 14 9 4 2
KO (player 4 shares a table with player 5 on round 2): 84 (once), 12 (twice)
(Current seating has a the same encounter repartition)
Rule 3 KO. Absolute deviation is: 0,182 => 1, 2, 6, 7, 8, 11, 12 have 15 VP | 3, 4, 5, 9, 10, 13, 14, 15, 16, 17, 18, 19 have 14 VP
(Current seating has the same absolute vp deviation)
(Current seating violates rule 7.)
Rule 8 KO. Absolute deviation is: 0,747 => 1, 7, 11, 17, 19 have 7 transfers | 2, 4, 5, 12, 13, 14, 15, 16, 18 have 9 transfers | 3, 6, 8, 9, 10 have 8 transfers
(Current seating has an absolute transfer deviation of 0,831)

20 players (3R) (in progress): 3 1 14 10 17 | 8 6 19 15 2 | 13 11 4 20 7 | 18 16 9 5 12 / 15 19 7 1 13 | 20 4 12 6 18 | 5 9 17 11 3 | 10 14 2 16 8
Rule 8. KO. Absolute deviation is: 0,72 => 1, 6, 11, 16 have 7 transfers | 2, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20 have 9 transfers | 3, 8, 13, 18 have 8 transfers

21 players (3R): to revalidate fixed in 1.5f
The seating in 1.5a has a better score on rule 2. "No pair of players share a table through all two rounds, when possible. (N/A in some 2R event.)" It is used until we found something better

Ok, we have a better seating for 15 players! Hooray!
If you see anything strange with this seating or have any comment, feel free to post!

In any case, it has a better transfer distribution, and more importantly, doesn't violate rules 6 and 7 anymore!

We also have a better seating for 14 players (better transfer distribution, and more importantly, it doesn't violate rule 7. "A player doesn't play in the same seat position, if possible." anymore!)